CSS code

In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of Stabilizer codes constructed from classical codes with some special properties.

Construction

Let C_1 and C_2 be two (classical)  [n,k_1],  [n,k_2] codes such, that  C_2 \subset C_1 and  C_1 , C_2^\perp both have minimal distance  \geq 2t+1, where  C_2^\perp is the code dual to  C_2. Then define  \text{CSS}(C_1,C_2), the CSS code of  C_1 over  C_2 as an  [n,k_1 - k_2, d] code, with  d \geq 2t+1 as follows:

Define for  x \in C_1 : {{|}} x + C_2 \rangle  :=  1 / \sqrt{ {{|}} C_2 {{|}} }  \sum_{y \in C_2} {{|}} x + y \rangle, where  + is bitwise addition modulo 2. Then  \text{CSS}(C_1,C_2) is defined as  \{ {{|}} x + C_2 \rangle \mid x \in C_1 \} .

References

    Michael A. Nielsen and Isaac L. Chuang (2010). "Quantum Computation and Quantum Information (10. Anniversary Edition)". Cambridge University Press.

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